International J.Math. Combin. Vol.1(2012), 106-112
The Line n-Sigraph of a Symmetric n-Sigraph-IV P. Siva Kota Reddy† , K. M. Nagaraja‡ and M. C. Geetha† †
‡
Department of Mathematics, Acharya Institute of Technology, Bangalore-560 090, India
Department of Mathematics, J S S Academy of Technical Education, Bangalore - 560 060, India E-mail: pskreddy@acharya.ac.in
Abstract: An n-tuple (a1 , a2 , · · · , an ) is symmetric, if ak = an−k+1 , 1 ≤ k ≤ n. Let Hn = {(a1 , a2 , · · · , an ) : ak ∈ {+, −}, ak = an−k+1 , 1 ≤ k ≤ n} be the set of all symmetric n-tuples. Asymmetric n-sigraph (symmetric n-marked graph) is an ordered pair Sn = (G, σ) (Sn = (G, µ)), where G = (V, E) is a graph called the underlying graph of Sn and σ : E → Hn (µ : V → Hn ) is a function. In Bagga et al. (1995) introduced the concept of the super line graph of index r of a graph G, denoted by Lr (G). The vertices of Lr (G) are the rsubsets of E(G) and two vertices P and Q are adjacent if there exist p ∈ P and q ∈ Q such that p and q are adjacent edges in G. Analogously, one can define the super line symmetric n-sigraph of index r of a symmetric n-sigraph Sn = (G, σ) as a symmetric nsigraph Lr (Sn ) = (Lr (G), σ ′ ), where Lr (G) is the underlying graph of Lr (Sn ), where for any edge P Q in Lr (Sn ), σ ′ (P Q) = σ(P )σ(Q). It is shown that for any symmetric nsigraph Sn , its Lr (Sn ) is i-balanced and we offer a structural characterization of super line symmetric n-sigraphs of index r. Further, we characterize symmetric n-sigraphs Sn for which Sn ∼ L2 (Sn ), L2 (Sn ) ∼ L(Sn ) and L2 (Sn ) ∼ Sn where ∼ denotes switching equivalence and L2 (Sn ), L(Sn ) and Sn are denotes the super line symmetric n-sigraph of index 2, line symmetric n-sigraph and complementary symmetric n-sigraph of Sn respectively. Also, we characterize symmetric n-sigraphs Sn for which Sn ∼ = L2 (Sn ) and L2 (Sn ) ∼ = L(Sn ).
Key Words: Smarandachely symmetric n-marked graph, symmetric n-sigraph, symmetric n-marked graph, balance, switching, balance, super line symmetric n-sigraph, line symmetric n-sigraph.
AMS(2010): 05C22 §1. Introduction Unless mentioned or defined otherwise, for all terminology and notion in graph theory the reader is refer to [6]. We consider only finite, simple graphs free from self-loops. Let n ≥ 1 be an integer. An n-tuple (a1 , a2 , · · · , an ) is symmetric, if ak = an−k+1 , 1 ≤ k ≤ n. Let Hn = {(a1 , a2 , · · · , an ) : ak ∈ {+, −}, ak = an−k+1 , 1 ≤ k ≤ n} be the set of all symmetric n-tuples. Note that Hn is a group under coordinate wise multiplication, and the 1 Received
September 26, 2011. Accepted March 16, 2012.
The Line n-Sigraph of a Symmetric n-Sigraph-IV
107
n order of Hn is 2m , where m = ⌈ ⌉. 2 A Smarandachely k-marked graph is an ordered pair S = (G, µ) where G = (V, E) is a graph called underlying graph of S and µ : V → (e1 , e2 , ..., ek ) is a function, where each ei ∈ {+, −}. An n-tuple (a1 , a2 , ..., an ) is symmetric, if ak = an−k+1 , 1 ≤ k ≤ n. Let Hn = {(a1 , a2 , ..., an ) : ak ∈ {+, −}, ak = an−k+1 , 1 ≤ k ≤ n} be the set of all symmetric n-tuples. A Smarandachely symmetric n-marked graph is an ordered pair Sn = (G, µ), where G = (V, E) is a graph called the underlying graph of Sn and µ : V → Hn is a function. Particularly, a Smarandachely 2-marked graph is called a symmetric n-sigraph (symmetric n-marked graph), where G = (V, E) is a graph called the underlying graph of Sn and σ : E → Hn (µ : V → Hn ) is a function. In this paper by an n-tuple/n-sigraph/n-marked graph we always mean a symmetric ntuple/symmetric n-sigraph/symmetric n-marked graph. An n-tuple (a1 , a2 , · · · , an ) is the identity n-tuple, if ak = +, for 1 ≤ k ≤ n, otherwise it is a non-identity n-tuple. In an n-sigraph Sn = (G, σ) an edge labelled with the identity n-tuple is called an identity edge, otherwise it is a non-identity edge. Further, in an n-sigraph Sn = (G, σ), for any A ⊆ E(G) the n-tuple σ(A) is the product of the n-tuples on the edges of A. In [12], the authors defined two notions of balance in n-sigraph Sn = (G, σ) as follows (See also R. Rangarajan and P.S.K.Reddy [9]: Definition 1.1 Let Sn = (G, σ) be an n-sigraph. Then, (i) Sn is identity balanced (or i-balanced), if product of n-tuples on each cycle of Sn is the identity n-tuple, and (ii) Sn is balanced, if every cycle in Sn contains an even number of non-identity edges. Note An i-balanced n-sigraph need not be balanced and conversely. The following characterization of i-balanced n-sigraphs is obtained in [12]. Proposition 1.1(E. Sampathkumar et al. [12]) An n-sigraph Sn = (G, σ) is i-balanced if, and only if, it is possible to assign n-tuples to its vertices such that the n-tuple of each edge uv is equal to the product of the n-tuples of u and v. In [12], the authors also have defined switching and cycle isomorphism of an n-sigraph Sn = (G, σ) as follows (See also [7,10,11] & [14]-[18]): Let Sn = (G, σ) and Sn′ = (G′ , σ ′ ), be two n-sigraphs. Then Sn and Sn′ are said to be isomorphic, if there exists an isomorphism φ : G → G′ such that if uv is an edge in Sn with label (a1 , a2 , · · · , an ) then φ(u)φ(v) is an edge in Sn′ with label (a1 , a2 , · · · , an ).
Given an n-marking µ of an n-sigraph Sn = (G, σ), switching Sn with respect to µ is the operation of changing the n-tuple of every edge uv of Sn by µ(u)σ(uv)µ(v). The n-sigraph obtained in this way is denoted by Sµ (Sn ) and is called the µ-switched n-sigraph or just switched n-sigraph. Further, an n-sigraph Sn switches to n-sigraph Sn′ (or that they are switching equivalent
108
P. Siva Kota Reddy, K. M. Nagaraja and M. C. Geetha
to each other), written as Sn ∼ Sn′ , whenever there exists an n-marking of Sn such that Sµ (Sn ) ∼ = Sn′ . Two n-sigraphs Sn = (G, σ) and Sn′ = (G′ , σ ′ ) are said to be cycle isomorphic, if there exists an isomorphism φ : G → G′ such that the n-tuple σ(C) of every cycle C in Sn equals to the n-tuple σ(φ(C)) in Sn′ . We make use of the following known result (see [12]).
Proposition 1.2(E. Sampathkumar et al. [12]) Given a graph G, any two n-sigraphs with G as underlying graph are switching equivalent if, and only if, they are cycle isomorphic. In this paper, we introduced the notion called super line n-sigraph of index r and we obtained some interesting results in the following sections. The super line n-sigraph of index r is the generalization of line n-sigraph.
§2. Super Line n-Sigraph Lr (Sn ) In [1], the authors introduced the concept of the super line graph, which generalizes the notion of line graph. For a given G, its super line graph Lr (G) of index r is the graph whose vertices are the r-subsets of E(G), and two vertices P and Q are adjacent if there exist p ∈ P and q ∈ Q such that p and q are adjacent edges in G. In [1], several properties of Lr (G) were studied. Many other properties and concepts related to super line graphs were presented in [2,4]. The study of super line graphs continues the tradition of investigating generalizations of line graphs in particular and of graph operators in general, as elaborated in the classical monograph by Prisner [8]. From the definition, it turns out that L1 (G) coincides with the line graph L(G). More specifically, some results regarding the super line graph of index 2 were presented in [3] and [5]. Several variations of the super line graph have been considered. In this paper, we extend the notion of Lr (G) to realm of n-sigraphs as follows: The super line n-sigraph of index r of an n-sigraph Sn = (G, σ) as an n-sigraph Lr (Sn ) = (Lr (G), σ ′ ), where Lr (G) is the underlying graph of Lr (Sn ), where for any edge P Q in Lr (Sn ), σ ′ (P Q) = σ(P )σ(Q). Hence, we shall call a given n-sigraph Sn a super line n-sigraph of index r if it is isomorphic to the super line n-sigraph of index r, Lr (Sn′ ) of some n-sigraph Sn′ . In the following subsection, we shall present a characterization of super line n-sigraph of index r. The following result indicates the limitations of the notion Lr (Sn ) as introduced above, since the entire class of i-unbalanced n-sigraphs is forbidden to be super line n-sigraphs of index r. Proposition 2.1 For any n-sigraph Sn = (G, σ), its Lr (Sn ) is i-balanced. ′
Proof Let σ denote the n-tuple of Lr (Sn ) and let the n-tuple σ of Sn be treated as an n-marking of the vertices of Lr (Sn ). Then by definition of Lr (Sn ) we see that σ ′ (P Q) = σ(P )σ(Q), for every edge P Q of Lr (Sn ) and hence, by Proposition 1.1, the result follows. Corollary 2.2 For any n-sigraph Sn = (G, σ), its L2 (Sn ) is i-balanced.
The Line n-Sigraph of a Symmetric n-Sigraph-IV
109
For any positive integer k, the k th iterated super line n-sigraph of index r, Lr (Sn ) of Sn is defined as follows: L0r (Sn ) = Sn , Lkr (Sn ) = Lr (Lk−1 (Sn )) r Corollary 2.3 For any n-sigraph Sn = (G, σ) and any positive integer k, Lkr (Sn ) is i-balanced. The line graph L(G) of graph G has the edges of G as the vertices and two vertices of L(G) are adjacent if the corresponding edges of G are adjacent. The line n-sigraph of an n-sigraph Sn = (G, σ) is an n-sigraph L(Sn ) = (L(G), σ ′ ), where for any edge ee′ in L(Sn ), σ ′ (ee′ ) = σ(e)σ(e′ ). This concept was introduced by E. Sampatkumar et al. [13]. The following result is one can easily deduce from Proposition 2.1. Corollary 2.4 (E. Sampathkumar et al. [13]) For any n-sigraph Sn = (G, σ), its line n-sigraph L(Sn ) is i-balanced. In [5], the authors characterized those graphs that are isomorphic to their corresponding super line graphs of index 2. Proposition 2.5(K. S. Bagga et al. [5]) For a graph G = (V, E), G ∼ = L2 (G) if, and only if, G = K3 . We now characterize the n-sigraphs that are switching equivalent to their super line nsigraphs of index 2. Proposition 2.6 For any n-sigraph Sn = (G, σ), Sn ∼ L2 (Sn ) if, and only if, G = K3 and S is i-balanced n-sigraph. Proof Suppose Sn ∼ L2 (Sn ). This implies, G ∼ = L2 (G) and hence G is K3 . Now, if Sn is any n-sigraph with underlying graph as K3 , Corollary 2.2 implies that L2 (Sn ) is i-balanced and hence if Sn is i-unbalanced and its L2 (Sn ) being i-balanced can not be switching equivalent to Sn in accordance with Proposition 1.2. Therefore, Sn must be i-balanced. Conversely, suppose that Sn is i-balanced n-sigraph and G is K3 . Then, since L2 (Sn ) is i-balanced as per Corollary 2.2 and since G ∼ = L2 (G), the result follows from Proposition 1.2 again. We now characterize the n-sigraphs that are isomorphic to their super line n-sigraphs of index 2. Proposition 2.7 For any n-sigraph Sn = (G, σ), Sn ∼ = L2 (Sn ) if, and only if, G = K3 and Sn is i-balanced n-sigraph. In [5], the authors characterized whose super line graphs of index 2 that are isomorphic to L(G). Proposition 2.8(K. S. Bagga et al. [5]) For a graph G = (V, E), L2 (G) ∼ = L(G) if, and only if, G is K1,3 , K3 or 3K2 .
110
P. Siva Kota Reddy, K. M. Nagaraja and M. C. Geetha
From the above result we have following result for signed graphs: Proposition 2.9 For any n-sigraph Sn = (G, σ), L2 (Sn ) ∼ L(Sn ) if, and only if, G is K1,3 , K3 or 3K2 . Proof Suppose L2 (Sn ) ∼ L(Sn ). This implies, L2 (G) ∼ = L(G) and hence by Proposition 2.8, we see that the graph G must be isomorphic to K1,3 , K3 or 3K2 . Conversely, suppose that G is a K1,3 , K3 or 3K2 . Then L2 (G) ∼ = L(G) by Proposition 2.8.
Now, if Sn any n-sigraph on any of these graphs, By Proposition 2.1 and Corollary 2.4, L2 (Sn ) and L(Sn ) are i-balanced and hence, the result follows from Proposition 1.2. We now characterize n-sigraphs whose super line n-sigraphs L2 (Sn ) that are isomorphic to line n-sigraphs. Proposition 2.10 For any n-sigraph Sn = (G, σ), L2 (Sn ) ∼ = L(Sn ) if, and only if, G is K1,3 , K3 or 3K2 .
Proof Clearly L2 (G) ∼ = L(G), when G is K1,3 , K3 or 3K2 . Consider the map f : V (L2 (G)) → V (L(G)) defined by f (e1 e2 , e2 e3 ) = (e1 , e3 ) is an isomorphism. Let σ be any n-tuple on K1,3 , K3 or 3K2 . Let e = (e1 e2 , e2 e3 ) be an edge in L2 (G), where G is K1,3 , K3 or 3K2 . Then the n-tuple of the edge e in L2 (G) is the σ(e1 e2 )σ(e2 e3 ) which is the n-tuple of the edge (e1 , e3 ) in L(G), where G is K1,3 , K3 or 3K2 . Hence the map f is also an n-sigraph isomorphism between L2 (Sn ) and L(Sn ). Let Sn = (G, σ) be an n-sigraph. The complement of Sn is an n-sigraph Sn = (G, σ c ), c where G is the underlying Ygraph of Sn and for any edge e = uv ∈ Sn , σ (uv) = µ(u)µ(v), where for any v ∈ V , µ(v) = σ(uv). Clearly, Sn as defined here is an i-balanced n-sigraph due u∈N (v)
to Proposition 1.1.
In [5], the authors proved there are no solutions to the equation L2 (G) ∼ G. So it is impossible to construct switching equivalence relation of L2 (Sn ) ∼ Sn for any arbitrary nsigraph. The following result characterizes n-sigraphs which are super line n-sigraphs of index r. Proposition 2.11 An n-sigraph Sn = (G, σ) is a super line n-sigraph of index r if and only if Sn is i-balanced n-sigraph and its underlying graph G is a super line graph of index r. Proof Suppose that Sn is i-balanced and G is a Lr (G). Then there exists a graph H such that Lr (H) ∼ = G. Since Sn is i-balanced, by Proposition 1.1, there exists an n-marking µ of G such that each edge uv in Sn satisfies σ(uv) = µ(u)µ(v). Now consider the n-sigraph Sn′ = (H, σ ′ ), where for any edge e in H, σ ′ (e) is the n-marking of the corresponding vertex in G. Then clearly, Lr (Sn′ ) ∼ = Sn . Hence Sn is a super line n-sigraph of index r. Conversely, suppose that Sn = (G, σ) is a super line n-sigraph of index r. Then there exists an n-sigraph Sn′ = (H, σ ′ ) such that Lr (Sn′ ) ∼ = Sn . Hence G is the Lr (G) of H and by Proposition 2.1, Sn is i-balanced.
The Line n-Sigraph of a Symmetric n-Sigraph-IV
111
If we take r = 1 in Lr (Sn ), then this is the ordinary line n-sigraph. In [13], the authors obtained structural characterization of line n-sigraphs and clearly Proposition 2.11 is the generalization of line n-sigraphs. Proposition 2.12(E. Sampathkumar et al. [13]) An n-sigraph Sn = (G, Ďƒ) is a line n-sigraph if, and only if, Sn is i-balanced n-sigraph and its underlying graph G is a line graph.
Acknowledgement The first and last authors are grateful to Sri. B. Premnath Reddy, Chairman, Acharya Institutes, for his constant support and encouragement for R & D.
References [1] K.S.Bagga, L.W. Beineke and B.N. Varma, Super line graphs, In: Y.Alavi, A.Schwenk (Eds.), Graph Theory, Combinatorics and Applications, vol. 1, Wiley-Interscience, New York, 1995, pp. 35-46. [2] K.S.Bagga, L.W.Beineke and B.N.Varma, The line completion number of a graph, In: Y.Alavi, A.Schwenk (Eds.), Graph Theory, Combinatorics and Applications, vol. 2, WileyInterscience, New York, 1995, pp. 1197-1201. [3] K.S.Bagga and M.R.Vasquez, The super line graph L2 (G) for hypercubes, Congr. Numer., 93 (1993), 111-113. [4] Jay Bagga, L.W.Beineke and B.N.Varma, Super line graphs and their properties, In: Y. Alavi, D.R. Lick, J.Q. Liu (Eds.), Combinatorics, Graph Theory, Algorithms and Applications, World Scientific, Singapore 1995. [5] K.S.Bagga, L.W.Beineke and B.N.Varma, The super line graph L2 (G), Discrete Math., 206 (1999), 51-61. [6] F.Harary, Graph Theory, Addison-Wesley Publishing Co., 1969. [7] V.Lokesha, P.S.K.Reddy and S.Vijay, The triangular line n-sigraph of a symmetric nsigraph, Advn. Stud. Contemp. Math., 19(1) (2009), 123-129. [8] E.Prisner, Graph Dynamics, Longman, London, 1995. [9] R.Rangarajan and P.Siva Kota Reddy, Notions of balance in symmetric n-sigraphs, Proceedings of the Jangjeon Math. Soc., 11(2) (2008), 145-151. [10] R.Rangarajan, P.S.K.Reddy and M.S.Subramanya, Switching Equivalence in Symmetric n-Sigraphs, Adv. Stud. Comtemp. Math., 18(1) (2009), 79-85. [11] R.Rangarajan, P.S.K.Reddy and N.D.Soner, Switching equivalence in symmetric n-sigraphsII, J. Orissa Math. Sco., 28 (1 & 2) (2009), 1-12. [12] E.Sampathkumar, P.S.K.Reddy, and M.S.Subramanya, Jump symmetric n-sigraph, Proceedings of the Jangjeon Math. Soc., 11(1) (2008), 89-95. [13] E.Sampathkumar, P.S.K.Reddy, and M.S.Subramanya, The Line n-sigraph of a symmetric n-sigraph, Southeast Asian Bull. Math., 34(5) (2010), 953-958.
112
P. Siva Kota Reddy, K. M. Nagaraja and M. C. Geetha
[14] P.S.K.Reddy and B.Prashanth, Switching equivalence in symmetric n-sigraphs-I, Advances and Applications in Discrete Mathematics, 4(1) (2009), 25-32. [15] P.S.K.Reddy, S.Vijay and B.Prashanth, The edge C4 n-sigraph of a symmetric n-sigraph, Int. Journal of Math. Sci. & Engg. Appls., 3(2) (2009), 21-27. [16] P.S.K.Reddy, V.Lokesha and Gurunath Rao Vaidya, The Line n-sigraph of a symmetric n-sigraph-II, Proceedings of the Jangjeon Math. Soc., 13(3) (2010), 305-312. [17] P.S.K.Reddy, V.Lokesha and Gurunath Rao Vaidya, The Line n-sigraph of a symmetric n-sigraph-III, Int. J. Open Problems in Computer Science and Mathematics, 3(5) (2010), 172-178. [18] P.S.K.Reddy, V.Lokesha and Gurunath Rao Vaidya, Switching equivalence in symmetric n-sigraphs-III, Int. Journal of Math. Sci. & Engg. Appls., 5(1) (2011), 95-101.